First, we estimate the Koopman operator \(K\) .
Let \([x_{1,0},\ldots ,x_{N,0}]=\mathbf {x}_0,\ldots ,[x_{1,T},\ldots ,x_{N,T}]=\mathbf {x}_T\in \mathcal {X}^N\)  be given data generated by the dynamical system (REF ) at time \(t_0,\ldots ,t_T\) .
We assume that \(S\)  sequences \(\lbrace \mathbf {x}^1_0,\ldots ,\mathbf {x}^1_T\rbrace ,\ldots ,\lbrace \mathbf {x}^S_0,\ldots ,\mathbf {x}^S_T\rbrace \)  of data are given.
For the Hilbert space \(\mathcal {H}\) , we use a vector-valued reproducing kernel Hilbert space  [1]}.
We generate a subspace of \(\mathcal {H}\)  by the given data and estimate the Koopman operator \(K\)  on the subspace.
The estimation of the Koopman operator is proposed in, e.g. [2]} and detailed in Section E.
